Fractions, Decimals, and Percents Arithmetic Practice Questions — GMAT

Question 1Question →

At an agricultural cooperative, $0.45$ of the total fruit harvested by weight consisted of apples, and $\frac{3}{8}$ of the remaining fruit weight consisted of pears. If the remaining $550$ kilograms of fruit consisted entirely of oranges, what was the total weight, in kilograms, of the fruit harvested?

Show answer & explanation

Answer: 1600

Answer

1600

Subtracting the apple portion leaves 55% (or 11/20) of the total harvest. Since pears make up 3/8 of this remainder, oranges make up the remaining 5/8 of the 11/20 portion, which simplifies to 11/32 of the total harvest. Setting 11/32 of the total harvest equal to 550 kg yields a total weight of 1,600 kg.

Step-by-Step Solution

1

Find the fraction of the total harvest left after subtracting the apples.

The remaining portion is 1 - 0.45 = 0.55, which equals 11/20 of the total harvest.

Apples account for 0.45 of the harvest.

2

Calculate the portion of the harvest represented by oranges.

Since pears are 3/8 of the remainder, oranges are 5/8 of the remainder. Thus, oranges are (5/8) * (11/20) = 11/32 of the total harvest.

The non-apple harvest consists only of pears and oranges.

3

Solve for the total weight using the given weight of oranges.

Total weight = 550 * (32 / 11) = 1600 kg.

11/32 of the total weight equals 550 kilograms.

Key Concept

Multi-step arithmetic combining decimals and fractions to solve remaining-quantity word problems

Question 2Question →

A consultancy allocated a total budget for a digital transformation project. In the first phase, $\frac{3}{8}$ of the initial budget was spent. In the second phase, $40\%$ of the remaining budget was spent. If the unspent amount after both phases is $\$27,000$ , what was the initial budget of the project, in dollars?

Show answer & explanation

Answer: 72000

Answer

72,000

To determine the initial budget, track the unspent fraction step by step. After the first phase,

\frac{5}{8}

of the budget remains. In the second phase,

40\%

(or

\frac{2}{5}

) of that remaining fraction is spent, which leaves

60\%

(or

\frac{3}{5}

) of the

\frac{5}{8}

unspent. Multiplying

\frac{3}{5} \times \frac{5}{8} = \frac{3}{8}

. Because

\frac{3}{8}

of the initial budget equals

\$27,000

, the full initial budget is

27,000 \times \frac{8}{3} = 72,000

.

Step-by-Step Solution

1

Find the fraction of the budget remaining after the first phase.

The fraction remaining after Phase 1 is

1 - \frac{3}{8} = \frac{5}{8}

.

The first phase spends

\frac{3}{8}

of the initial total budget.

2

Find the fraction of the total initial budget spent in the second phase.

Phase 2 expenditure =

40\% \times \frac{5}{8} = \frac{2}{5} \times \frac{5}{8} = \frac{2}{8}

.

40\%

is equivalent to

\frac{2}{5}

, which applies to the remaining

\frac{5}{8}

of the budget.

3

Determine the remaining fraction of the total budget after both phases.

Remaining fraction =

\frac{5}{8} - \frac{2}{8} = \frac{3}{8}

.

Subtracting the fraction spent in Phase 2 from the fraction left after Phase 1 yields the unspent portion.

4

Calculate the initial total budget in dollars.

Initial Budget =

\$27,000 \times \frac{8}{3} = \$72,000

.

Setting

\frac{3}{8}

of the total budget equal to the unspent amount of

\$27,000

determines the total.

Key Concept

Sequential fractional and percentage reductions from a base quantity

Question 3Question →

A clothing store reduced the original price of a jacket by $20\%$ . During a promotional event, the store offered an additional $15\%$ discount off the reduced price. If the final price of the jacket was $\$102$ , what was the original price of the jacket, in dollars?

Show answer & explanation

Answer: 150

Answer

The original price of the jacket was $150.

To find the original price, express the final price as a decimal multiplier of the original price. A

20\%

discount leaves

80\%

of the original price (

0.80

). An additional

15\%

discount off the reduced price leaves

85\%

of that amount (

0.85

). The overall price multiplier is

0.80 \times 0.85 = 0.68

. Setting

0.68 \times \text{Original Price} = 102

gives an original price of

\frac{102}{0.68} = 150

dollars.

Step-by-Step Solution

1

Determine the combined price multiplier after both successive discounts.

The first discount leaves

80\%

(

0.80

) of the original price, and the second discount leaves

85\%

(

0.85

) of that reduced price. The combined multiplier is

0.80 \times 0.85 = 0.68

.

Successive percentage discounts are multiplicative rather than additive.

2

Set up an equation equating the final discounted price to

102

.

0.68 \times P = 102

, where

P

represents the original price.

The final price is

68\%

of the original base price.

3

Solve for the original price

P

.

P = \frac{102}{0.68} = \frac{10200}{68} = 150

.

Dividing the final amount by the net decimal multiplier yields the original value.

Key Concept

Successive percentage change calculation

Estimated Time:1m 0s

Question 4Question →

A wholesale distributor imported a batch of organic green tea leaves. On Monday, $30\%$ of the initial batch was sold. On Tuesday, $\frac{4}{7}$ of the remaining batch was sold. If the distributor had $126$ kilograms of green tea leaves left at the end of Tuesday, what was the total weight, in kilograms, of the initial batch imported?

294

315

420

700

980

Show answer & explanation

Answer: $420$

Answer

The total weight of the initial batch imported was

420

kilograms.

Selling

30\%

on Monday leaves

70\%

of the initial total. On Tuesday, selling

\frac{4}{7}

of that remaining amount leaves

\frac{3}{7}

of the

70\%

. Calculating

\frac{3}{7} \times 70\% = 30\%

. Since

30\%

of the initial batch is equal to

126

kg, the total initial weight is

126 \div 0.30 = 420

kg.

Step-by-Step Solution

1

Determine the fraction of the initial batch remaining after Monday's sale.

Since

30\%

(

0.30

) was sold,

1 - 0.30 = 0.70

(or

\frac{7}{10}

) of the initial batch remained.

Percentage decreases are calculated relative to the original whole.

2

Determine the fraction of Monday's remaining batch that was left after Tuesday's sale.

Since

\frac{4}{7}

of Monday's remainder was sold,

1 - \frac{4}{7} = \frac{3}{7}

of Monday's remainder was left.

Subsequent fractional sales are relative to the updated intermediate amount.

3

Calculate the final remaining amount as a fraction of the initial total weight

W

.

\text{Final Remaining} = \frac{3}{7} \times \left(\frac{7}{10} W\right) = \frac{3}{10} W = 0.30 W$.

Multiplying successive remaining ratios gives the net remaining fraction of the original batch.

4

Solve for the initial weight

W

using the given remaining weight of

126

kg.

0.30 W = 126 \implies W = \frac{126}{0.30} = 420\text{ kg}.

Dividing the remaining weight by its corresponding decimal fraction yields the total initial weight.

Key Concept

Successive percentage and fractional reductions require applying each change to the updated intermediate base value rather than the original total.

Estimated Time:1m 30s

Question 5Question →

At the beginning of the year, a logistics company's fleet consisted of trucks, vans, and cargo planes. Exactly $0.40$ of the total fleet were trucks, and $\frac{3}{8}$ of the total fleet were vans, with the remaining vehicles being cargo planes. Over the course of the year, the number of trucks increased by $25\%$ , the number of vans decreased by $20\%$ , and the number of cargo planes increased by $50\%$ . By what percentage did the total number of vehicles in the company's fleet increase over the year?

11.25\%

13.75\%

15.00\%

18.33\%

55.00\%

Show answer & explanation

Answer: $13.75\%$

Answer

The total number of vehicles in the fleet increased by

13.75\%

.

To find the net percentage increase of the entire fleet, we express each vehicle subgroup as a fraction or decimal of the initial total fleet

T

. Trucks account for

0.40T

, vans account for

\frac{3}{8}T = 0.375T

, and cargo planes account for the remaining

1 - (0.40 + 0.375) = 0.225T

. Applying the respective percentage changes gives an increase of

0.25 \times 0.40T = +0.10T

for trucks, a decrease of

0.20 \times 0.375T = -0.075T

for vans, and an increase of

0.50 \times 0.225T = +0.1125T

for cargo planes. Summing these changes yields a net gain of

0.10T - 0.075T + 0.1125T = 0.1375T

, which corresponds to an overall increase of

13.75\%

.

Step-by-Step Solution

1

Determine the initial proportions of each vehicle type in the fleet.

Trucks represent

0.40

of the fleet. Vans represent

\frac{3}{8} = 0.375

of the fleet. Cargo planes represent

1 - (0.40 + 0.375) = 0.225

of the fleet.

The sum of all component proportions must equal

1.0

(or

100\%

of the initial fleet).

2

Calculate the weighted net change contributed by each vehicle type relative to the initial fleet size

T

.

Truck change:

+25\% \text{ of } 0.40T = 0.25 \times 0.40T = +0.100T

.
Van change:

-20\% \text{ of } 0.375T = -0.20 \times 0.375T = -0.075T

.
Cargo plane change:

+50\% \text{ of } 0.225T = 0.50 \times 0.225T = +0.1125T

.

The net change contributed by a subgroup is its relative proportion multiplied by its specific percentage change.

3

Sum the component changes to find the total overall change.

Total net change

= +0.100T - 0.075T + 0.1125T = +0.1375T

.

Combining the individual net contributions yields the overall change in fleet size.

4

Convert the decimal net change to a percentage.

0.1375 \times 100\% = 13.75\%

.

Multiplying the decimal fraction of total increase by

100\%

gives the overall percentage increase.

Key Concept

Weighted Percentage Changes

Alternative Method

Assume a concrete total initial fleet size that works easily with the numbers, such as

T = 800

vehicles. Initial trucks

= 0.40 \times 800 = 320

. Initial vans

= \frac{3}{8} \times 800 = 300

. Initial planes

= 800 - 320 - 300 = 180

. After changes: new trucks

= 320 \times 1.25 = 400

, new vans

= 300 \times 0.80 = 240

, new planes

= 180 \times 1.50 = 270

. New total fleet

= 400 + 240 + 270 = 910

. Fleet increase

= 910 - 800 = 110

. Percentage increase

= \frac{110}{800} \times 100\% = 13.75\%

.

Estimated Time:2m 0s

Question 6Question →

At a logistics distribution hub, a shipment of incoming packages was processed over two shifts. During the morning shift, $0.35$ of the total shipment was processed and dispatched. During the evening shift, $\frac{4}{13}$ of the remaining packages were processed. If $450$ packages remained unprocessed at the end of both shifts, what was the total number of packages in the initial shipment?

Show answer & explanation

Answer: 1000

Answer

1000 packages

To find the initial shipment size, first express

0.35

as the fraction

\frac{7}{20}

. Subtracting this from

1

leaves

\frac{13}{20}

of the initial total after the morning shift. Next, processing

\frac{4}{13}

of these remaining packages leaves

1 - \frac{4}{13} = \frac{9}{13}

of that remaining amount. Multiplying

\frac{9}{13} \times \frac{13}{20}

shows that

\frac{9}{20}

of the original total remains unprocessed. Finally, setting

\frac{9}{20}

of the total equal to

450

packages yields an initial total of

1000

packages.

Step-by-Step Solution

1

Convert decimal portion to fraction and find remaining fraction after morning shift

Remaining fraction after morning shift is

\frac{13}{20}

Since

0.35 = \frac{7}{20}

of the total shipment was processed,

1 - \frac{7}{20} = \frac{13}{20}

of the initial shipment remained.

2

Determine the remaining fraction of the shipment after the evening shift

Final remaining fraction of the initial shipment is

\frac{9}{20}

The evening shift processed

\frac{4}{13}

of the remaining packages, leaving

1 - \frac{4}{13} = \frac{9}{13}

of the remaining packages. Thus,

\frac{9}{13} \times \frac{13}{20} = \frac{9}{20}

of the initial shipment remained.

3

Solve for the total initial number of packages

Total initial packages =

1000

Setting

\frac{9}{20} N = 450

gives

N = 450 \times \frac{20}{9} = 1000

.

Key Concept

Combining decimal-to-fraction conversions with successive remaining fraction calculations

Question 7Question →

A commercial bakery uses flour, sugar, and butter as the main ingredients by weight to produce a specialized pastry mix. Flour accounts for $0.50$ of the total weight of the mix. Sugar accounts for $\frac{3}{8}$ of the remaining weight of the mix. If the rest of the mix consists of $15$ kilograms of butter, what is the total weight, in kilograms, of the pastry mix?

24

48

60

80

120

Show answer & explanation

Answer: $48$

Answer

The total weight of the pastry mix is

48

kilograms.

The option stating

48

is correct because flour leaves

\frac{1}{2}

of the total mix. Sugar takes

\frac{3}{8}

of that half, leaving

\frac{5}{8}

of the half for butter. Thus, butter accounts for

\frac{5}{16}

of the total mix. Solving

\frac{5}{16} \times W = 15

yields

W = 48

kg.

Step-by-Step Solution

1

Determine the fraction of the total weight remaining after accounting for flour.

Since flour represents

0.50 = \frac{1}{2}

of the total weight

W

, the remaining weight is

1 - \frac{1}{2} = \frac{1}{2}W

.

The base for the sugar component is specified as the remaining weight after flour.

2

Calculate the fraction of the total weight represented by butter.

Sugar takes

\frac{3}{8}

of the remaining weight, so butter takes the remaining

1 - \frac{3}{8} = \frac{5}{8}

of the remaining weight. Therefore, butter is

\frac{5}{8} \times \frac{1}{2}W = \frac{5}{16}W

.

Butter forms the rest of the mixture after sugar is accounted for within the remaining weight.

3

Set up an equation with the given butter weight to solve for total weight

W

.

\frac{5}{16}W = 15 \implies W = 15 \times \frac{16}{5} = 3 \times 16 = 48$ kg.

Equating the algebraic fractional expression to the known numerical weight yields the overall total.

Key Concept

Multi-step successive fraction arithmetic with changing base values

Question 8Question →

At the start of a month, an online bookstore's inventory consisted of Fiction, Non-Fiction, and Textbook titles. Exactly $\frac{3}{8}$ of the total inventory consisted of Fiction titles, and $0.40$ of the remaining inventory consisted of Non-Fiction titles, with the balance consisting of Textbook titles. During the month, the number of Fiction titles increased by $20\%$ , the number of Non-Fiction titles decreased by $25\%$ , and the number of Textbook titles remained unchanged. If the total inventory increased by a net amount of $15$ titles at the end of the month, how many total book titles were in the inventory at the start of the month?

400

600

800

1,200

2,400

Show answer & explanation

Answer: 1,200

Answer

1,200 total book titles

The initial inventory consists of

\frac{3}{8}T

Fiction titles and

0.40 \times \frac{5}{8}T = \frac{1}{4}T

Non-Fiction titles. A

20\%

increase in Fiction adds

\frac{3}{40}T

, while a

25\%

decrease in Non-Fiction subtracts

\frac{1}{16}T

. Combining these gives a net change of

\frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = \frac{1}{80}T

. Setting

\frac{1}{80}T = 15

gives

T = 1,200

.

Step-by-Step Solution

1

Determine the initial fractional breakdown of each book category relative to the total inventory T

Fiction

= \frac{3}{8}T

. Remaining inventory

= 1 - \frac{3}{8} = \frac{5}{8}T

. Non-Fiction

= 0.40 \times \frac{5}{8}T = \frac{2}{5} \times \frac{5}{8}T = \frac{1}{4}T

.

The question specifies that Non-Fiction is 0.40 of the remaining inventory, not the total inventory.

2

Calculate the net change in titles as a fraction of total initial inventory T

Increase in Fiction

= 20\% \times \frac{3}{8}T = \frac{1}{5} \times \frac{3}{8}T = +\frac{3}{40}T

.
Decrease in Non-Fiction

= 25\% \times \frac{1}{4}T = \frac{1}{4} \times \frac{1}{4}T = -\frac{1}{16}T

.
Net fractional change

= \frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = +\frac{1}{80}T

.

Percentage changes must be applied to each category's specific share of the total inventory.

3

Equate the net fractional change to the numerical net increase and solve for T

\frac{1}{80}T = 15 \implies T = 15 \times 80 = 1,200.

The overall net increase is given as 15 titles.

Key Concept

Multi-step successive fraction and percentage change with changing base values

Estimated Time:2m 0s

Question 9Question →

At the beginning of a fiscal year, a municipal transit authority allocated its capital expenditure budget among three projects: Bus Rapid Transit, Rail Modernization, and Station Upgrades. Exactly $0.30$ of the total budget was allocated to Bus Rapid Transit. Of the remaining budget, exactly $\frac{3}{7}$ was allocated to Rail Modernization, and the rest was allocated to Station Upgrades. By the end of the year, expenditures on Bus Rapid Transit exceeded its initial allocation by $25\%$ , expenditures on Rail Modernization were $15\%$ below its initial allocation, and expenditures on Station Upgrades exceeded its initial allocation by $10\%$ . By what percent did the transit authority's total expenditures across all three projects exceed its initial total budget?

Show answer & explanation

Answer: 7

Answer

The total expenditures across all three projects exceeded the initial total budget by

7\%

.

To solve this problem, represent the total initial budget as

B

. The Bus Rapid Transit allocation is

0.30B

, leaving

0.70B

. Rail Modernization receives

\frac{3}{7}

of

0.70B

, which equals

0.30B

. The remaining portion for Station Upgrades is

0.70B - 0.30B = 0.40B

. End-of-year expenditures are calculated by multiplying each allocation by its respective growth multiplier: Bus Rapid Transit is

0.30B \times 1.25 = 0.375B

, Rail Modernization is

0.30B \times 0.85 = 0.255B

, and Station Upgrades is

0.40B \times 1.10 = 0.44B

. Summing these expenditures gives

0.375B + 0.255B + 0.44B = 1.07B

. Comparing

1.07B

to the initial

1.00B

reveals an overall increase of

0.07B

, or

7\%

.

Step-by-Step Solution

1

Express the initial allocations for each project as fractions of the total budget B

Bus Rapid Transit =

0.30B

, Rail Modernization =

0.30B

, Station Upgrades =

0.40B

Bus Rapid Transit is explicitly

0.30B

. The remaining

0.70B

is split such that Rail Modernization receives

\frac{3}{7} \times 0.70B = 0.30B

, leaving

0.70B - 0.30B = 0.40B

for Station Upgrades.

2

Apply the individual percentage changes to determine end-of-year expenditures

Bus Rapid Transit =

0.375B

, Rail Modernization =

0.255B

, Station Upgrades =

0.44B

A

25\%

increase corresponds to a multiplier of

1.25

, a

15\%

decrease corresponds to a multiplier of

0.85

, and a

10\%

increase corresponds to a multiplier of

1.10

.

3

Sum the project expenditures and calculate the net percent change relative to B

Total expenditure =

1.07B

, corresponding to a

7\%

net increase

Adding

0.375B + 0.255B + 0.44B

yields

1.07B

. Subtracting the original budget

1.00B

gives

0.07B

, which is

7\%

of

B

.

Key Concept

Weighted Percentage Changes and Sequential Fraction-Decimal Operations

Question 10Question →

In January, electricity comprised $0.45$ of a facility's total energy consumption. In February, electricity comprised $\frac{3}{5}$ of its total energy consumption. If the facility's total energy consumption was constant at $1,200$ kilowatt-hours in each of the two months, by how many kilowatt-hours did its electricity consumption increase from January to February?

150

kilowatt-hours

180

kilowatt-hours

270

kilowatt-hours

540

kilowatt-hours

720

kilowatt-hours

Show answer & explanation

Answer: $180$ kilowatt-hours

Answer

The electricity consumption increased by

180

kilowatt-hours from January to February.

The correct answer of

180

kilowatt-hours is found by converting both fractions and decimals into comparable parts of the

1,200

kilowatt-hour base. January electricity usage is

0.45 \times 1,200 = 540

kWh. February electricity usage is

\frac{3}{5} \times 1,200 = 720

kWh. The difference is

720 - 540 = 180

kWh. Alternatively, subtracting the proportions first gives

\frac{3}{5} - 0.45 = 0.60 - 0.45 = 0.15

, and

0.15 \times 1,200 = 180

kWh.

Step-by-Step Solution

1

Calculate the January electricity consumption

0.45 \times 1,200 = 540

kilowatt-hours

Electricity represented

0.45

of the

1,200

kilowatt-hour total.

2

Calculate the February electricity consumption

\frac{3}{5} \times 1,200 = 0.60 \times 1,200 = 720

kilowatt-hours

Electricity represented

\frac{3}{5}

(or

0.60

) of the

1,200

kilowatt-hour total.

3

Find the difference between February and January consumption

720 - 540 = 180

kilowatt-hours

Subtracting January usage from February usage yields the net increase.

Key Concept

Fraction and Decimal Conversions and Amount Computations

Question 11Question →

In 2025, an architectural design firm divided its working hours among commercial, residential, and municipal projects. Commercial projects accounted for $0.40$ of the total hours. Of the remaining hours, $\frac{5}{12}$ were spent on residential projects, and the rest were spent on municipal projects.

In 2026, the firm's total working hours increased by $25\%$ . Commercial project hours decreased by $15\%$ , while residential project hours increased by $40\%$ . By what percent did the number of hours spent on municipal projects increase from 2025 to 2026?

Show answer & explanation

Answer: 60

Answer

The number of hours spent on municipal projects increased by

60\%

.

By setting the total initial hours to a variable

T

(or a convenient constant like

100

), we find that in 2025, commercial projects comprised

0.40T

, residential comprised

\frac{5}{12} \times 0.60T = 0.25T

, and municipal comprised

0.35T

. In 2026, total hours rose to

1.25T

, commercial dropped to

0.34T

, and residential rose to

0.35T

, leaving

0.56T

for municipal. The percent increase in municipal hours is

\frac{0.56T - 0.35T}{0.35T} = \frac{0.21}{0.35} = 60\%

.

Step-by-Step Solution

1

Determine the baseline breakdown of working hours for 2025 in terms of total hours

T

.

Commercial hours =

0.40T

, Residential hours =

0.25T

, Municipal hours =

0.35T

.

Commercial is given as

0.40T

. The remaining fraction

0.60T

is split such that

\frac{5}{12} \times 0.60T = 0.25T

goes to residential, leaving

0.60T - 0.25T = 0.35T

for municipal.

2

Calculate the updated working hours for each category in 2026.

Total hours =

1.25T

, Commercial hours =

0.34T

, Residential hours =

0.35T

.

A

25\%

total increase yields

1.25T

. A

15\%

decrease in commercial hours yields

0.40T \times 0.85 = 0.34T

. A

40\%

increase in residential hours yields

0.25T \times 1.40 = 0.35T

.

3

Find the municipal project hours for 2026 by subtracting commercial and residential hours from total 2026 hours.

Municipal hours (2026) =

0.56T

.

Municipal hours in 2026 equal

1.25T - 0.34T - 0.35T = 0.56T

.

4

Compute the percent change in municipal project hours from 2025 to 2026.

Percent Increase =

60\%

.

Percent increase is

\frac{\text{Municipal}_{2026} - \text{Municipal}_{2025}}{\text{Municipal}_{2025}} \times 100\% = \frac{0.56T - 0.35T}{0.35T} \times 100\% = \frac{0.21}{0.35} \times 100\% = 60\%

.

Key Concept

Multi-step percentage change and fraction-decimal conversions

Estimated Time:2m 0s

Question 12Question →

At the start of a fiscal year, a sovereign wealth fund allocated its capital into three asset classes: Stocks, Bonds, and Commodities. Exactly $0.40$ of the total capital was allocated to Stocks. Of the remaining capital, $\frac{1}{3}$ was allocated to Bonds and the rest was allocated to Commodities. Over the course of the fiscal year, the value of Stocks increased by $15\%$ and the value of Bonds decreased by $10\%$ . If the total value of the fund increased by $5\%$ by the end of the fiscal year, by what percent did the value of Commodities change?

Show answer & explanation

Answer: 2.5

Answer

The value of Commodities increased by

2.5\%

.

By setting the initial total capital to

V

, the initial values are Stocks =

0.40V

, Bonds =

\frac{1}{3}(0.60V) = 0.20V

, and Commodities =

0.60V - 0.20V = 0.40V

. Applying the percentage changes gives final values of Stocks =

0.46V

, Bonds =

0.18V

, and Total Fund =

1.05V

. The final Commodities value is

1.05V - 0.64V = 0.41V

. The percent change in Commodities is

\frac{0.41V - 0.40V}{0.40V} \times 100\% = 2.5\%

.

Step-by-Step Solution

1

Determine initial asset allocations as proportions of total capital

V

Stocks (

S_0

) =

0.40V

, Bonds (

B_0

) =

0.20V

, Commodities (

C_0

) =

0.40V

Stocks take

0.40V

, leaving

0.60V

. Bonds take

\frac{1}{3}

of

0.60V = 0.20V

, leaving

0.40V

for Commodities.

2

Compute final values for Stocks, Bonds, and Total Fund after percentage changes

S_1 = 0.46V

,

B_1 = 0.18V

, and Total Fund

V_1 = 1.05V

Stocks increase by

15\%

(

0.40V \times 1.15 = 0.46V

), Bonds decrease by

10\%

(

0.20V \times 0.90 = 0.18V

), and the total fund grows by

5\%

(

1.05V

).

3

Calculate the end-of-year value of Commodities

C_1 = 0.41V

Subtract final Stocks and Bonds from total fund:

C_1 = 1.05V - (0.46V + 0.18V) = 0.41V

.

4

Calculate the percent change of Commodities relative to its initial value

2.5\%

increase

Percent change

= \frac{0.41V - 0.40V}{0.40V} \times 100\% = \frac{0.01}{0.40} \times 100\% = 2.5\%

.

Key Concept

Weighted percentage change and fractional portion modeling

Estimated Time:1m 40s

Question 13Question →

A commercial coffee roastery prepares a batch of specialty coffee beans containing Arabica, Robusta, and Liberica beans. Initially, Arabica beans make up $0.50$ of the total weight of the batch, Robusta beans make up $\frac{3}{10}$ of the total weight, and Liberica beans make up the remaining weight. After a specialized roasting process, the weight of the Arabica beans decreases by $10\%$ , the weight of the Robusta beans decreases by $50\%$ , and the weight of the Liberica beans remains unchanged. What percent of the final total weight of the coffee batch is made up of Liberica beans?

Show answer & explanation

Answer: 25

Answer

25%

To find the percentage of Liberica beans in the final mixture, first determine the initial weight breakdown assuming a total weight of 100 units: Arabica is 50 units, Robusta is 30 units, and Liberica is 20 units. After roasting, Arabica weight decreases by 10% to 45 units, Robusta weight decreases by 50% to 15 units, and Liberica weight stays at 20 units. The new total weight is 45 + 15 + 20 = 80 units. The proportion of Liberica in the final batch is 20 out of 80 units, which equals 1/4 or 25%.

Step-by-Step Solution

1

Find the initial fractional and percentage composition of the coffee blend.

Arabica accounts for 50% (0.50), Robusta accounts for 30% (3/10), and Liberica accounts for 20% (1 - 0.50 - 0.30 = 0.20).

The sum of all components in the initial blend must equal 1 (or 100%).

2

Assume an initial reference weight of 100 units to represent the batch.

Initial Arabica weight = 50 units, initial Robusta weight = 30 units, and initial Liberica weight = 20 units.

Using a convenient base value like 100 simplifies multi-step percentage change calculations without loss of generality.

3

Calculate the post-roasting weight for each component.

New Arabica weight = 50 * (1 - 0.10) = 45 units. New Robusta weight = 30 * (1 - 0.50) = 15 units. New Liberica weight = 20 units.

Apply the respective percentage decreases to each individual component weight.

4

Determine the new total weight of the batch.

Total final weight = 45 + 15 + 20 = 80 units.

The final total weight is the sum of the remaining weights of all three bean types.

5

Compute the final percentage of Liberica beans.

(20 / 80) * 100% = 25%.

Divide the final weight of Liberica beans by the final total weight of the batch and convert to a percentage.

Key Concept

Combining initial fractions and percentages to compute multi-step composition changes

Question 14Question →

The original price of a sweater is $\$40$ . If the price is increased by $25\%$ , and then the new price is decreased by $10\%$ , what is the final price of the sweater?

\$44

\$45

\$46

\$50

\$36

Show answer & explanation

Answer: $\$45$

Answer

The final price of the sweater is

\$45

.

Increasing the original price of

\$40

by

25\%

yields an intermediate price of

\$50

(since

25\%

of

\$40

is

\$10

). Decreasing

\$50

by

10\%

reduces the price by

\$5

(since

10\%

of

\$50

is

\$5

), resulting in a final price of

\$45

.

Step-by-Step Solution

1

Calculate the price after the initial

25\%

increase.

The price increases by

25\%

of

\$40

, which is

0.25 \times \$40 = \$10

. The new intermediate price is

\$40 + \$10 = \$50

.

Percentage increases must be calculated using the starting base value.

2

Calculate the final price after the

10\%

decrease.

The discount is

10\%

of the intermediate price of

\$50

, which is

0.10 \times \$50 = \$5

. Subtracting this from the intermediate price gives

\$50 - \$5 = \$45

.

Successive percentage changes apply to the updated base value, not the original starting value.

Key Concept

Successive Percentage Changes and Base Value Alignment

Question 15Question →

A retail warehouse received a large shipment of electronics. On Monday, the warehouse sold $\frac{2}{5}$ of the total shipment. On Tuesday, it sold $25\%$ of the items that remained after Monday. On Wednesday, it sold $33\frac{1}{3}\%$ of the items that remained after Tuesday. If $120$ items remained unsold at the end of Wednesday, how many items were in the original shipment?

400

600

1,200

7,200

36

Show answer & explanation

Answer: $400$

Answer

The original shipment contained

400

items.

The correct option correctly evaluates the fraction remaining after each successive sale. After selling

\frac{2}{5}

on Monday,

\frac{3}{5}

remained. After selling

25\%

(

\frac{1}{4}

) on Tuesday,

\frac{3}{4}

of the previous remainder (

\frac{9}{20}

) remained. After selling

33\frac{1}{3}\%

(

\frac{1}{3}

) on Wednesday,

\frac{2}{3}

of that remainder remained, giving a net remaining fraction of

\frac{3}{10}

. Equating

\frac{3}{10}

of the total to

120

yields an original shipment size of

400

.

Step-by-Step Solution

1

Calculate the fraction of items remaining after Monday's sales.

Since

\frac{2}{5}

were sold,

1 - \frac{2}{5} = \frac{3}{5}

of the original shipment remained.

The remaining fraction is

1

minus the fraction sold.

2

Calculate the fraction of items remaining after Tuesday's sales.

Tuesday sold

25\% = \frac{1}{4}

of the remaining items, leaving

1 - \frac{1}{4} = \frac{3}{4}

of Monday's remainder. The fraction remaining relative to the original total is

\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}

.

Successive percentage reductions apply to the updated intermediate remainder.

3

Calculate the fraction of items remaining after Wednesday's sales.

Wednesday sold

33\frac{1}{3}\% = \frac{1}{3}

of Tuesday's remainder, leaving

1 - \frac{1}{3} = \frac{2}{3}

of that remainder. The final fraction remaining relative to the original total is

\frac{9}{20} \times \frac{2}{3} = \frac{6}{20} = \frac{3}{10}

.

Multiply by the fraction remaining after Wednesday's reduction.

4

Set up the equation to solve for the original total number of items

N

.

\frac{3}{10}N = 120 \implies N = 120 \times \frac{10}{3} = 400

.

Divide the remaining count by the net remaining fraction.

Key Concept

Successive Percent and Fraction Reductions

Estimated Time:1m 30s

Question 16Question →

A store owner purchases a batch of books for $\$150$ each and sells them at a price that is $40\%$ higher than the purchase price. During a clearance sale, the selling price is reduced by $10\%$ . What is the final sale price, in dollars, of one book?

Show answer & explanation

Answer: 189

Answer

The final sale price of one book is

\$189

.

First, calculate the price after the

40\%

markup:

\$150 \times 1.40 = \$210

. Then apply the

10\%

discount to this new amount:

\$210 \times 0.90 = \$189

. The final price is

\$189

.

Step-by-Step Solution

1

Calculate the price after a 40% markup on the base price of $150.

$210

A 40% markup means multiplying the original purchase price by 1.40.

2

Apply the 10% clearance discount to the marked-up price of $210.

$189

A 10% discount means multiplying the new price by 0.90.

Key Concept

Successive Percentage Change

Estimated Time:45s

Question 17Question →

At a technology firm, $\frac{3}{8}$ of the total annual operating budget was initially allocated to Research & Development, $0.35$ of the budget was allocated to Marketing, and the remainder was allocated to Operations. Mid-year, the Marketing allocation was increased by $20\%$ of its initial value, and the Operations allocation was decreased by $40\%$ of its initial value, while the Research & Development allocation remained unchanged. By what net percentage did the firm's total annual operating budget change?

A decrease of

20\%

A decrease of

6\%

A decrease of

4\%

An increase of

4\%

A decrease of

15\%

Show answer & explanation

Answer: A decrease of $4\%$

Answer

A net decrease of

4\%

Converting

\frac{3}{8}

to

37.5\%

leaves

27.5\%

for Operations (

100\% - 37.5\% - 35\%

). A

20\%

increase on the

35\%

Marketing budget adds

7\%

to the overall total (

0.20 \times 35\% = +7\%

). A

40\%

decrease on the

27.5\%

Operations budget reduces the overall total by

11\%

(

0.40 \times 27.5\% = -11\%

). Summing these changes gives

+7\% - 11\% = -4\%

, representing a net decrease of

4\%

.

Step-by-Step Solution

1

Convert initial fractional and decimal allocations to percentages of the total budget

Research & Development allocation =

\frac{3}{8} = 37.5\%

; Marketing allocation =

0.35 = 35\%

Converting all components to a common percentage format enables straightforward comparison and arithmetic.

2

Calculate the initial percentage allocated to Operations

Operations allocation =

100\% - (37.5\% + 35\%) = 100\% - 72.5\% = 27.5\%

The remainder of the budget after R&D and Marketing constitutes the Operations department share.

3

Determine the net percentage change contributed by each department's adjustment

Marketing change =

+20\% \text{ of } 35\% = +7\%

of total budget; Operations change =

-40\% \text{ of } 27.5\% = -11\%

of total budget; R&D change =

0\%

Each departmental percentage change must be weighted by that department's portion of the overall budget.

4

Sum the net contributions to find the overall budget change

Net overall change =

+7\% - 11\% = -4\%

(a decrease of

4\%

)

Combining the positive and negative adjustments yields the overall net percentage change relative to the initial budget.

Key Concept

Weighted Percent Change across Mixed Fractional and Decimal Sub-components

Question 18Question →

At a biotechnology facility, a solution tank initially contains a mixture of ethanol, water, and stabilizer liquid. By volume, $0.36$ of the initial mixture is ethanol and $\frac{2}{5}$ is water, with the remainder consisting of stabilizer liquid. During a purification process, $25\%$ of the ethanol and $\frac{1}{8}$ of the water evaporate, while the volume of the stabilizer liquid remains unchanged. If the volume of the remaining stabilizer liquid is $132$ liters, by what percentage did the total volume of the liquid mixture in the tank decrease?

14%

16.28%

18.75%

37.50%

24.00%

Show answer & explanation

Answer: 14%

Answer

The total volume of the liquid mixture in the tank decreased by 14%.

The initial mixture consists of 36% ethanol, 40% water, and 24% stabilizer by volume. Ethanol loses 25% of its volume, which corresponds to

0.25 \times 36\% = 9\%

of the total initial mixture volume. Water loses

\frac{1}{8}

of its volume, corresponding to

\frac{1}{8} \times 40\% = 5\%

of the total initial mixture volume. Combining these losses gives

9\% + 5\% = 14\%

net decrease relative to the initial total volume.

Step-by-Step Solution

1

Express initial component proportions as unified decimals or fractions.

Ethanol fraction =

0.36 = \frac{9}{25}

. Water fraction =

\frac{2}{5} = 0.40 = \frac{10}{25}

. Stabilizer fraction =

1 - (0.36 + 0.40) = 0.24 = \frac{6}{25}

.

Establishing all initial parts relative to the total initial volume allows step-by-step proportion tracking.

2

Calculate the fraction of total volume lost due to evaporation for each component.

Ethanol loss =

25\% \times 0.36 = 0.25 \times 0.36 = 0.09

of initial total volume. Water loss =

\frac{1}{8} \times 0.40 = 0.125 \times 0.40 = 0.05

of initial total volume.

Applying the component-specific percentage decrease to each component's fraction yields its contribution to the overall volume loss.

3

Sum the component volume losses to find the total percentage decrease.

Total volume loss fraction =

0.09 + 0.05 = 0.14

, which equals

14\%

of the total initial volume.

The stabilizer experience no loss, so the net change in total volume is the direct sum of the losses in ethanol and water relative to the original base.

Key Concept

Multi-step arithmetic operations combining decimals, fractions, and weighted percent changes using a consistent base value.

Estimated Time:1m 50s

Question 19Question →

A cloud data center processes an incoming raw data stream through sequential filtering stages. In Stage 1, $\frac{3}{8}$ of the incoming raw data volume is discarded as noise, and $0.20$ of the remaining data is flagged for long-term archiving. In Stage 2, $\frac{4}{5}$ of the data not flagged for long-term archiving is processed into active storage, while the rest is discarded. If the volume of data processed into active storage in Stage 2 is $50$ terabytes greater than the volume of data discarded in Stage 1, what was the initial volume of the raw data stream, in terabytes?

400

1,600

2,000

2,500

4,000

Show answer & explanation

Answer: 2,000

Answer

2,000 terabytes

Let the initial data volume be

X

. Discarded in Stage 1 is

\frac{3}{8}X

, leaving

\frac{5}{8}X

. Archiving takes

20\%

(

0.20 = \frac{1}{5}

) of this remainder, which is

\frac{1}{5} \times \frac{5}{8}X = \frac{1}{8}X

. The data entering Stage 2 is

\frac{5}{8}X - \frac{1}{8}X = \frac{1}{2}X

. Active storage in Stage 2 is

\frac{4}{5} \times \frac{1}{2}X = \frac{2}{5}X

. Subtracting Stage 1 discarded volume from Stage 2 active storage volume gives

\frac{2}{5}X - \frac{3}{8}X = \frac{16}{40}X - \frac{15}{40}X = \frac{1}{40}X

. Setting

\frac{1}{40}X = 50

yields

X = 2,000

terabytes.

Step-by-Step Solution

1

Define the variable and compute Stage 1 discarded volume and remaining volume.

Discarded in Stage 1 =

\frac{3}{8}X

; Remaining after Stage 1 discard =

X - \frac{3}{8}X = \frac{5}{8}X

.

Establishing quantities in terms of the total initial volume

X

allows setting up a single-variable linear equation.

2

Calculate the volume flagged for archiving and the volume available for Stage 2.

Archived volume =

0.20 \times \frac{5}{8}X = \frac{1}{5} \times \frac{5}{8}X = \frac{1}{8}X

. Available for Stage 2 = \frac{5}{8}X - \frac{1}{8}X = \frac{4}{8}X = \frac{1}{2}X$.

The

0.20

decimal must be converted to a fraction (

\frac{1}{5}

) and applied to the remaining

\frac{5}{8}X

base.

3

Calculate the volume processed into active storage in Stage 2.

Active storage volume =

\frac{4}{5} \times \frac{1}{2}X = \frac{2}{5}X

.

Stage 2 processes

\frac{4}{5}

of the data that entered Stage 2.

4

Set up the algebraic equation comparing active storage volume in Stage 2 to discarded volume in Stage 1.

\frac{2}{5}X - \frac{3}{8}X = 50

.

The problem states that active storage volume in Stage 2 is

50

terabytes greater than the Stage 1 discarded volume.

5

Find a common denominator and solve for

X

.

\left(\frac{16}{40} - \frac{15}{40}\right)X = 50 \implies \frac{1}{40}X = 50 \implies X = 2,000

.

Converting fractions to a denominator of

40

yields

\frac{1}{40}X = 50

, so multiplying by

40

gives

X = 2,000

terabytes.

Key Concept

Multi-step arithmetic operations involving sequential fractions, decimals, and percent bases

Estimated Time:2m 0s

Question 20Question →

At a chemical refining plant, a raw liquid compound containing Substance X, Substance Y, and an inert solvent is processed in two sequential purification phases. Initially, Substance X accounts for $0.25$ of the total weight of the compound, and Substance Y accounts for $\frac{2}{5}$ of the remaining weight, with the inert solvent comprising the rest. In Phase 1, $20\%$ of Substance X and $30\%$ of Substance Y are removed, while all of the inert solvent is retained. In Phase 2, a certain percentage $p\%$ of the inert solvent present after Phase 1 is removed, while no other substances are removed. If Substance X represents exactly $40\%$ of the total weight of the compound remaining after Phase 2, what is the value of $p$ ?

Show answer & explanation

Answer: 80

Answer

The value of

p

is 80.

Assuming a total initial weight of 100 units, Substance X is 25 units and the remaining weight is 75 units. Substance Y is

\frac{2}{5}

of 75, which equals 30 units, leaving 45 units of inert solvent. After Phase 1, 20 units of Substance X and 21 units of Substance Y remain, along with the full 45 units of solvent. In Phase 2, Substance X (20 units) becomes

40\%

of the total mixture, making the final total weight

\frac{20}{0.40} = 50

units. Since Substance X and Substance Y together account for

20 + 21 = 41

units, the remaining solvent after Phase 2 must be

50 - 41 = 9

units. Reducing solvent from 45 units down to 9 units requires removing

\frac{45 - 9}{45} = \frac{36}{45} = 80\%

of the solvent. Thus,

p = 80

.

Step-by-Step Solution

1

Determine initial component amounts using decimal and fractional breakdown.

In a 100-unit mixture, Substance X = 25 units, Substance Y = 30 units, and Inert Solvent = 45 units.

Substance X is

0.25

of the total (

25

units). Of the remaining

75

units, Substance Y is

\frac{2}{5} \times 75 = 30

units. The rest (

75 - 30 = 45

units) is inert solvent.

2

Calculate remaining component amounts after Phase 1 percentage reductions.

Substance X = 20 units, Substance Y = 21 units, Inert Solvent = 45 units.

Removing

20\%

of Substance X leaves

25 \times 0.80 = 20

units. Removing

30\%

of Substance Y leaves

30 \times 0.70 = 21

units. No solvent is removed in Phase 1.

3

Determine the final total mixture weight using the final percentage of Substance X.

Final total mixture weight = 50 units.

Substance X (20 units) represents

40\%

(

0.40

) of the final mixture after Phase 2, so the total weight is

\frac{20}{0.40} = 50

units.

4

Formulate and solve the linear equation for

p

.

p = 80

.

The total weight is the sum of all remaining components:

20 + 21 + 45\left(1 - \frac{p}{100}\right) = 50

. Solving

41 + 45\left(1 - \frac{p}{100}\right) = 50

yields

45\left(1 - \frac{p}{100}\right) = 9

, so

1 - \frac{p}{100} = 0.20

, giving

p = 80

.

Key Concept

Multi-step percentage change, fractional remaining parts, and algebraic mixture equations

Fractions, Decimals, and Percents Arithmetic

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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